Building on recent advances in the analysis and design of algebraic flux correction (AFC) schemes, new differentiable limiter functions are constructed for efficient nonlinear solution strategies. The proposed scaling parameters are used to limit artificial diffusion operators incorporated into the residual of a high order target scheme to produce accurate and bound-preserving finite element approximations to hyperbolic problems. Due to this stabilization procedure, the occurring system becomes highly nonlinear and the efficient computation of corresponding solutions is a challenging task. The presented regularization approach makes the AFC residual twice continuously differentiable so that Newton’s method converges quadratically for sufficiently good initial guesses. Furthermore, the performance of each nonlinear iteration is improved by expressing the Jacobian as the sum and product of matrices having the same sparsity pattern as the Galerkin system matrix. Eventually, the AFC methodology constructed is validated numerically by applying it to several numerical benchmarks.

基于代数通量校正（AFC）方案的分析和设计的最新进展，构造了新的可微分限制器函数，以实现有效的非线性求解策略。拟议的缩放参数用于限制人工扩散算子并入高阶目标方案的残差中，以产生双曲线问题的精确且保留边界的有限元逼近。由于这种稳定程序，出现的系统变得高度非线性，并且相应解的有效计算是一项艰巨的任务。提出的正则化方法使AFC残差两次连续可微，因此Newton方法二次收敛以求足够好的初始猜测。此外，通过将雅可比行列式表示为与Galerkin系统矩阵具有相同稀疏度的矩阵的和和乘积，可以改善每个非线性迭代的性能。最终，通过将AFC方法应用于几个数字基准，对其进行了数字验证。